Moh Shaw-Kwei. Ming-t'i yian-suan ti kung-li hsi-t'ung (Some axiom systems for propositional calculus). Acta mathematica Sinica, vol. 5 (1955), pp. 117–135. English summary reprinted in Zentralblatt für Mathematik und ihre Grenzgebiete, vol. 64 (1957), p. 11.

1960 ◽  
Vol 25 (2) ◽  
pp. 182-183
Author(s):  
Hao Wang
1968 ◽  
Vol 33 (1) ◽  
pp. 27-38 ◽  
Author(s):  
R. A. Bull

In [2] Prior puts forward a tense logic, GH1, which is intended to axiomatise tense logic with time linear and rational; he also contemplates the tense logic with time linear and real. The purpose of this paper is to give completeness proofs for three axiom systems, GH1, GHlr, GHli, with respect to tense logic with time linear and rational, real, and integral, respectively.1 In a fourth section I show that GH1 and GHlr have the finite model property, but that GHli lacks it.GH1 has the operators of the classical propositional calculus, together with operators P, H, F, G for ‘It has been the case that’, ‘It has always been the case that’, ‘It will be the case that’, ‘It will always be the case that’, respectively.


1955 ◽  
Vol 20 (02) ◽  
pp. 109-114 ◽  
Author(s):  
Bolesław Sobociński

In this Journal, vol. 18 (1953), p. 350 (Problem 7), Prof. P. Bernays proposed the following problem on propositional calculus: What is the smallest number n such that the propositional calculus, formulated with substitution and modus ponens as the only rules of inference, can be based on a set of initial formulas each of which contains at most n propositional letters (counted with multiplicity) ? In this note I give a solution to this problem, viz., that this number n = 5. For a system of propositional calculus in which the primitive functors are “C” (implication) and “N” (negation) and in which there are only two rules of inference, i.e. the rules of substitution and detachment (modus ponens), the following can be proved. (1) A set of propositional theses each of which contains at most 4 propositional letters is inadequate to give the complete bi-valued calculus of propositions. (2) There are axiom systems for this calculus in which each axiom contains at most 5 propositional letters. § 1. Consider the following normal metrix, in which the designated value is I: This satisfies the two rules of inference, and the following. (a) The law of commutation, i.e. the thesis CCpCqrCqCpr. (b) The following theses: Furthermore, in this matrix “N” is defined in such a way that: (c) For any well-formed formula α and any value m of this matrix, α = m if and only if NNα = m.


Author(s):  
Julian Oliver Dörr ◽  
Georg Licht ◽  
Simona Murmann

AbstractCOVID-19 placed a special role on fiscal policy in rescuing companies short of liquidity from insolvency. In the first months of the crisis, SMEs as the backbone of Germany’s economy benefited from large and mainly indiscriminate aid measures. Avoiding business failures in a whatever-it-takes fashion contrasts, however, with the cleansing mechanism of economic crises: a mechanism which forces unviable firms out of the market, thereby reallocating resources efficiently. By focusing on firms’ pre-crisis financial standing, we estimate the extent to which the policy response induced an insolvency gap and analyze whether the gap is characterized by firms which were already struggling before the pandemic. With the policy measures being focused on smaller firms, we also examine whether this insolvency gap differs with respect to firm size. Our results show that the COVID-19 policy response in Germany has triggered a backlog of insolvencies that is particularly pronounced among financially weak, small firms, having potential long-term implications on entrepreneurship and economic recovery.Plain English Summary This study analyzes the extent to which the strong policy support to companies in the early phase of the COVID-19 crisis has prevented a large wave of corporate insolvencies. Using data of about 1.5 million German companies, it is shown that it was mainly smaller firms that experienced strong financial distress and would have gone bankrupt without policy assistance. In times of crises, insolvencies usually allow for a reallocation of employees and capital to more efficient firms. However, the analysis reveals that this ‘cleansing effect’ is hampered in the current crisis as the largely indiscriminate granting of liquidity subsidies and the temporary suspension of the duty to file for insolvency have caused an insolvency gap that is driven by firms which were already in a weak financial position before the crisis. Overall, the insolvency gap is estimated to affect around 25,000 companies, a substantial number compared to the around 16,300 actual insolvencies in 2020. In the ongoing crisis, policy makers should prefer instruments favoring entrepreneurs who respond innovatively to the pandemic instead of prolonging the survival of near-insolvent firms.


1984 ◽  
Vol 7 (3) ◽  
pp. 357-358
Author(s):  
Regimantas Pliuškevičius
Keyword(s):  

A Gentzen-style axiomatization of dynamic logics is proposed. It is compared to other axiom systems for dynamic logics. Some problems of relative decidability are considered.


2020 ◽  
Vol 6 (1) ◽  
Author(s):  
Femke van Schelven ◽  
Eline van der Meulen ◽  
Noortje Kroeze ◽  
Marjolijn Ketelaar ◽  
Hennie Boeije

Plain English summary Background Young people with a chronic condition are increasingly involved in doing research and developing tools and interventions that concern them. Working together with patients is called Patient and Public Involvement (PPI). We know from the literature that PPI with young people with a chronic condition can be challenging. Therefore, it is important that everyone shares their lessons learned from doing PPI. Aim We want to share our lessons learned from a large program, called Care and Future Prospects. This program helps young people with a chronic condition to, for example, go to school or to find a job. It funded numerous projects that could contribute to this. In all projects, project teams collaborated with young people with a chronic condition. What did we do We asked young people with a chronic condition and project teams about their experiences with PPI. Project teams wrote reports, were interviewed, and filled out a tool called the Involvement Matrix. Young people filled out a questionnaire. Findings In the article, we present our lessons learned. Examples are: it is important to involve young people with a chronic condition from the start of a project and everyone involved in a project should continuously discuss their responsibilities. We provide practical tips on how young people with a chronic condition and project teams can do this. A tip for young people is, for example: ‘discuss with the project team what you can and want to do and what you need’. An example of a tip for project teams is: ‘Take time to listen attentively to the ideas of young people’. Abstract Background The Patient and Public Involvement (PPI) of young people with a chronic condition receives increasing attention in policy and practice. This is, however, not without its challenges. Consequently, calls have been made to share lessons learned during PPI practice. Methods We share our lessons learned from a large participatory program, called Care and Future Prospects. This program aims to improve the social position of young people aged 0–25 with a physical or mental chronic condition by funding participatory projects. We have drawn our lessons from 33 of these projects, using four data sources. One data source provided information from the perspective of young people with a chronic condition, i.e. questionnaires. Three data sources contained information from the perspectives of project teams, i.e. project reports, case studies of projects and Involvement Matrices. For most of the projects, we have information from multiple data sources. Results We have combined the findings derived from all four data sources. This resulted in multiple lessons learned about PPI with young people with a chronic condition. Those lessons are divided into six themes, including practicalities to take into account at the start, involvement from the start, roles and responsibilities, support, flexibility and an open mind, and evaluation of process and outcomes. Conclusions The lessons learned have taught us that meaningful PPI requires effort, time and resources from both young people and project teams, from the beginning to the end. It is important to continuously discuss roles and responsibilities, and whether these still meet everyone’s needs and wishes. Our study adds to previous research by providing practical examples of encountered challenges and how to deal with them. Moreover, the practical tips can be a valuable aid by showing young people and project teams what concrete actions can support a successful PPI process.


Author(s):  
Alex Stewart

AbstractSome scholars assert that entrepreneurship has attained “considerable” legitimacy. Others assert that it “is still fighting” for complete acceptance. This study explores the question, extrapolating from studies of an “elite effect” in which the publications of the highest ranked schools differ from other research-intensive schools. The most elite business schools in the USA, but not the UK, are found to allocate significantly more publications to mathematically sophisticated “analytical” fields such as economics and finance, rather than entrepreneurship and other “managerial” fields. The US elites do not look down upon entrepreneurship as such. They look down upon journals that lack high mathematics content. Leading entrepreneurship journals, except Small Business Economics Journal (SBEJ), are particularly lacking. The conclusion argues that SBEJ can help the field’s legitimacy, but that other journals should not imitate analytical paradigms.Plain English Summary Academic snobs shun entrepreneurship journals. A goal for snobs is to exhibit superiority over others. For business professors, one way to do this is with mathematically sophisticated, analytical publications. Entrepreneurship journals, Small Business Economics excepted, do this relatively infrequently. These journals focus on the lives, activities, and challenges of diverse entrepreneurs. In the USA, the most elite business schools, compared with not-quite elite business schools, allocate significantly more of their articles to the journals of analytical fields such as economics, and fewer to entrepreneurship journals. This pattern is not found in the UK, where elites may have other ways to signal superiority. These elites, who accommodate entrepreneurship researchers, could pioneer with outputs of both relevance and scholarly quality, through collaboration between their practice-based and research-based professors.


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